Different from the traditional semi-supervised learning paradigm that is constrained by the close-world assumption, Generalized Category Discovery (GCD) presumes that the unlabeled dataset contains new categories not appearing in the labeled set, and aims to not only classify old categories but also discover new categories in the unlabeled data. Existing studies on GCD typically devote to transferring the general knowledge from the self-supervised pretrained model to the target GCD task via some fine-tuning strategies, such as partial tuning and prompt learning. Nevertheless, these fine-tuning methods fail to make a sound balance between the generalization capacity of pretrained backbone and the adaptability to the GCD task. To fill this gap, in this paper, we propose a novel adapter-tuning-based method named AdaptGCD, which is the first work to introduce the adapter tuning into the GCD task and provides some key insights expected to enlighten future research. Furthermore, considering the discrepancy of supervision information between the old and new classes, a multi-expert adapter structure equipped with a route assignment constraint is elaborately devised, such that the data from old and new classes are separated into different expert groups. Extensive experiments are conducted on 7 widely-used datasets. The remarkable improvements in performance highlight the effectiveness of our proposals.